Optimality Conditions for Generalized Ky Fan Quasi-Inequalities with Applications

In this paper, the image space analysis is employed to study a generalized Ky Fan quasi-inequality with cone constraints. By virtue of a nonlinear scalarization function and a positive linear operator, a nonlinear (regular) weak separation function and a linear regular weak separation function are introduced. Nonlinear and, in particular, linear separations for the generalized Ky Fan quasi-inequality with cone constraints are characterized. Some necessary and sufficient optimality conditions, especially a saddle-point sufficient optimality condition for the generalized Ky Fan quasi-inequality with cone constraints, are obtained. As applications, some sufficient conditions for (weak) vector equilibrium flows of vector traffic equilibrium problems with capacity arc constraints, are derived.     

Second-order Optimality Conditions for Mathematical Program with Semidefinite Cone Complementarity Constraints and Applications

Set-Valued and Variational Analysis - This paper is concerned with second-order optimality conditions for the mathematical program with semidefinite cone complementarity constraints. To achieve...     

Necessary Optimality Conditions for Mathematical Programs with Second-Order Cone Complementarity Constraints

This paper considers the mathematical program with second-order cone complementarity constrains (MPSOCC). As a generalization of the developed mathematical program with complementarity constrains (MPCC), MPSOCC has many applications in practice. Motivated by the MPCC theory, several stationarity concepts, which include the Clarke-type, Mordukhovich-type, and strong stationarities, are presented in this paper. It is further shown that a local minimizer of MPSOCC must be stationary in some sense under suitable conditions. This indicates that these stationarity concepts are reasonable in theory.     

锥约束非光滑多目标优化问题的对偶及最优性条件

研究了一类涉广义不变凸锥约束非光滑多目标优化问题(记为(MOP)),结合Craven与Yang广义选择定理,建立了该优化问题的Kuhn-Tucker型最优性充分必要条件以及其鞍点与弱有效解之间的关系,给出了(MOP)的Wolfe型与Mond-Weir型弱、强以及逆对偶理论.     

Necessary Optimality Conditions for Problems with Equality and Inequality Constraints: Abnormal Case

This article concerns second-order necessary conditions for an abnormal local minimizer of a nonlinear optimization problem with equality and inequality constraints. The obtained optimality conditions improve the ones available in the literature in that the associated set of Lagrange multipliers is the smallest possible. The first and the second authors were supported by Russian Foundation of Basic Research, Projects 08-01-90267, 08-01-90001. The second and third authors were supported by FCT (Portugal), Research Projects SFRH/BPD/26231/2006, PTDC/EEA-ACR/75242/2006.     

加权Fan Ky不等式及其加细

本文简证了加权Ｋｙ Ｆａｎ不等式，给出了两种加细形式。     

具有等式与不等式约束条件拟可微优化的Lagrange乘子型最优性条件

讨论了不等式约束优化问题中拟微分形式下Ｆｒｉｔｚ Ｊｏｈｎ必要条件与 Ｃｌａｒｋｅ广义梯度形式下Ｆｒｉｔｚ Ｊｏｈｎ必要条件的关系．在较弱条件下给出了具有等式与不等式约束条件的两个Ｌａｇｒａｎｇｅ乘子形式的最优性必要条件,在这两个条件中等式约束函数的拟微分和Ｃｌａｒｋｅ广义梯度分别被使用。     

Mathematical Programs with Semidefinite Cone Complementarity Constraints: Constraint Qualifications and Optimality Conditions

The tangent cone of gph $N_{S^n_+}$ plays an important role in developing necessary conditions for mathematical programs with semidefinite cone complementarity constraints. We demonstrate an elegant formula for the tangent cone of gph $N_{S^n_+}$ , based on which the Bouligand stationary point is characterized explicitly. The relationships among different stationary points under certain constraint qualifications are discussed. Then we propose a second order sufficient condition which can be weakened under the strict complementarity condition. Importantly, for the sake of algorithm design, under the assumption of strict complementarity condition, we give a nonsmooth equation reformulation of the stationary point, whose smoothing system is verified to be nonsingular at the stationary point under the proposed second order sufficient condition.     

Optimality Conditions for Minimax Optimization Problems with an Infinite Number of Constraints and Related Applications

This paper is concerned with the study of optimality conditions for minimax optimization problems with an infinite number of constraints,denoted by(MMOP).More precisely,we first establish necessary conditions for optimal solutions to the problem(MMOP)by means of employing some advanced tools of variational analysis and generalized differentiation.Then,sufficient conditions for the existence of such solutions to the problem(MMOP)are investigated with the help of generalized convexity functions defined in terms of the limiting subdifferential of locally Lipschitz functions.Finally,some of the obtained results are applied to formulating optimality conditions for weakly efficient solutions to a related multiobjective optimization problem with an infinite number of constraints,and a necessary optimality condition for a quasiε-solution to problem(MMOP).     

On Some Generalizations and Refinements of a Ky Fan Inequality

In this paper, a Ky Fan inequality and an inequality by Wu and Wang [10] will be generalized. Some new and improved refinements of the Ky Fan inequality will be put forward.AMS Subject Classification (2000) 26D15     

Second-Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints

We study second-order optimality conditions for mathematical programs with equilibrium constraints (MPEC). Firstly, we improve some second-order optimality conditions for standard nonlinear programming problems using some newly discovered constraint qualifications in the literature, and apply them to MPEC. Then, we introduce some MPEC variants of these new constraint qualifications, which are all weaker than the MPEC linear independence constraint qualification, and derive several second-order optimality conditions for MPEC under the new MPEC constraint qualifications. Finally, we discuss the isolatedness of local minimizers for MPEC under very weak conditions.     

Sufficient Conditions for Global Optimality of Bivalent Nonconvex Quadratic Programs with Inequality Constraints

We present sufficient conditions for the global optimality of bivalent nonconvex quadratic programs involving quadratic inequality constraints as well as equality constraints. By employing the Lagrangian function, we extend the global subdifferential approach, developed recently in Jeyakumar et al. (J. Glob. Optim., 2007, to appear; Math. Program. Ser. A, 2007, to appear) for studying bivalent quadratic programs without quadratic constraints, and derive global optimality conditions. The authors are grateful to the referees for constructive comments and suggestions which have contributed to the final preparation of the paper. Z.Y. Wu’s current address: School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat, Victoria, Australia. The work of this author was completed while at the Department of Applied Mathematics, University of New South Wales, Sydney, Australia.     

二阶锥约束随机变分不等式问题的数值方法研究

本文研究二阶锥约束随机变分不等式(SOCCSVI)问题,运用样本均值近似(SAA)方法结合光滑Fischer-Burmeister互补函数来求解该问题.首先,将SOCCSVI问题的Karush-Kuhn-Tucker系统转化为与之等价的方程组,并证明了该方程组的雅可比矩阵的非奇异性.其次,构造了光滑牛顿算法求解该方程组.最后,文章给出了两个数值实验证明了算法的有效性.     

Feasible Perturbations of Control Systems with Pure State Constraints and Applications to Second-Order Optimality Conditions

We propose second-order necessary optimality conditions for optimal control problems with very general state and control constraints which hold true under weak regularity assumptions on the data. In particular the pure state constraints are general closed sets, the optimal control is supposed to be merely measurable and the dynamics may be discontinuous in the time variable as well. These results are obtained by an approach based on local perturbations of the reference process by second-order tangent directions. This method allows direct and quite simple proofs.     

Sufficient Optimality Conditions for Optimal Control Problems with State Constraints

It is well-known in optimal control theory that the maximum principle, in general, furnishes only necessary optimality conditions for an admissible process to be an optimal one. It is also well-known that if a process satisfies the maximum principle in a problem with convex data, the maximum principle turns to be likewise a sufficient condition. Here an invexity type condition for state constrained optimal control problems is defined and shown to be a sufficient optimality condition. Further, it is demonstrated that all optimal control problems where all extremal processes are optimal necessarily obey this invexity condition. Thus optimal control problems which satisfy such a condition constitute the most general class of problems where the maximum principle becomes automatically a set of sufficient optimality conditions.     

Optimality Conditions and Exact Penalty for Mathematical Programs with Switching Constraints

Journal of Optimization Theory and Applications - In this paper, we mainly study the Abadie constraint qualification (ACQ) and the strong ACQ of a convex multifunction. To characterize the general...     

Optimality Conditions for Linear-Convex Optimal Control Problems with Mixed Constraints

Journal of Optimization Theory and Applications - In this paper, we provide sufficient optimality conditions for convex optimal control problems with mixed constraints. On one hand, the data...     

Ky Fan极大极小不等式在H-空间的进一步推广和应用

本文在Ｈ－空间得到一个新的ｍｉｎｉｍａｘ不等式,并得到了与其等价的几何形式、极大元存在定理及作为其应用的不动点定理。